Define the inverse secant function by restricting the domain of the secant function to the intervals (0, pi/2) and (pi/2, pi), and sketch the inverse function’s graph.

sec = 1/cos

note cos is 0 at pi/2 so sec is undefined there (infinite + or -)
in quadrant I cos is + so sec is +
but in quadrant II cos is - so sec is -

note cos always </= |1| so |sec| always >/= |1|

that should help with your graph

The inverse secant function, denoted as arcsec(x) or sec^(-1)(x), is the inverse of the restricted secant function with the domain restricted to the intervals (0, pi/2) and (pi/2, pi).

To define the inverse secant function, we need to consider the domain and range of the restricted secant function. The secant function generally has a domain of all real numbers except where cosine is zero, i.e., where the angle is (2n + 1)π/2 for any integer n. However, we restrict the domain to (0, pi/2) and (pi/2, pi) to ensure that the inverse secant function is one-to-one and has a well-defined inverse.

The restricted secant function is defined as:

sec(x) = 1/cos(x)

Since the secant function is not one-to-one on its entire domain, we restrict it to the intervals (0, pi/2) and (pi/2, pi) where it is one-to-one. In these intervals, the function has a restricted range of all positive real numbers.

To sketch the graph of the inverse secant function, we start by reflecting the graph of the restricted secant function about the line y = x since the inverse function is the reflection of the original function.

Since the restricted secant function has a domain of (0, pi/2) and (pi/2, pi) and a range of all positive real numbers, the inverse secant function will have a domain of all positive numbers and a range of (0, pi/2) and (pi/2, pi).

Plotting points on the graph, we find that arcsec(1) = 0, arcsec(2) ≈ 1.047, arcsec(3) ≈ 1.231, etc. As x approaches infinity, arcsec(x) approaches pi/2. Similarly, as x approaches negative infinity, arcsec(x) approaches pi/2.

Connecting these points gives a curve that starts at (1, 0) in the first quadrant, approaches the asymptote y = pi/2 as x approaches infinity, and approaches the asymptote y = 0 as x approaches negative infinity. The portion of the graph in the second quadrant follows a similar pattern with the asymptote y = pi in the first quadrant.

This is a rough visualization of the graph of the inverse secant function. Keep in mind that the scales of the x-axis and y-axis may vary depending on the software or graphing tool you are using.